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Theoretical and Mathematical Physics

Some thoughts on a nonlinear Schrödinger equation motivated by information theory

Rajesh R. Parwani

Published Jul 1, 2007• 5 citations
Publisher
DOI
10.1007/s11232-007-0085-1

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I begin by reviewing the arguments leading to a nonlinear generalisation of Schrodinger's equation within the context of the maximum uncertainty principle. Some exact and perturbative properties of that equation are then summarised: those results depend on a free regulating/interpolation parameter ηηη. I discuss here how one may fix that parameter using energetics. Other issues discussed are, a linear theory with an external potential that reproduces some unusual exact solutions of the nonlinear equation, and possible symmetry enhancements in the nonlinear theory.
We update our understanding of nonlinear Schrödinger equations motivated through information theory. In particular we show that a q —deformation of the basic nonlinear equation leads to a perturbative increase in the energy of a system, thus favouring the simplest q = 1 case. Furthermore the energy minimisation criterion is shown to be equivalent, at leading order, to an uncertainty maximisation argument. The special value η = 1/4 for the interpolation parameter, where leading order energy shifts vanish, implies the preservation of existing supersymmetry in nonlinearised supersymmetric quantum mechanics. Physically, η might be encoding relativistic effects.
Using a set of general methods developed by Krotov [A. I. Konnov and V. A. Krotov, Automation and Remote Control 60, 1427 (1999)], we extend the capabilities of optimal control theory to the nonlinear Schr\"odinger equation (NLSE). The paper begins with a general review of the Krotov approach to optimization. Although the linearized version of the method is sufficient for the linear Schr\"odinger equation, the full flexibility of the general method is required for treatment of the nonlinear Schr\"odinger equation. Formal equations for the optimization of the NLSE, as well as a concrete algorithm are presented. As an illustration, we consider a Bose-Einstein condensate (BEC) initially at rest in a harmonic trap. A phase develops across the BEC when an optical lattice potential is turned on. The goal is to counter this effect and keep the phase flat by adjusting the trap strength. The problem is formulated in the language of optimal control theory (OCT) and solved using the above methodology. To our knowledge, this is the first rigorous application of OCT to the nonlinear Schr\"odinger equation, a capability that is bound to have numerous other applications.